Averaged Extensions of Golomb's Triangular Recursion: Critical Invariance and Supercritical Constraints

Abstract

For an integer $m\ge 1$ and a parameter $\hat{I}\pm >0$, consider the nested recursion $$Q_{\hat{I}\pm ,m}(n)=1+\lfloor \frac{\hat{I}}{\pm }m\sum _{j=1}^m{Q}_{\hat{I}\pm ,m}\left(n-Q_{\hat{I}\pm ,m}(n-j)\right)\rfloor ,n>m,$$ with $Q_{\hat{I}\pm ,m}(1)=\cdots =Q_{\hat{I}\pm ,m}(m)=1$. For $m=1$ and $\hat{I}\pm =1$, this is Golomb's non-homogeneous triangular recursion. We prove that its canonical triangular solution is preserved, up to an initial index shift, by every finite arithmetic averaging length. More generally, the same exact solution is generated by any aggregator satisfying a local floor-lock condition. This class includes all power means of finite order, including the harmonic and geometric means, as well as the minimum and positively weighted quasi-arithmetic means. For $m\ge 2$, the maximum lies outside this class but has a different explicit block law. Consequently, every value $k\ge 2$ occurs exactly $k$ times in the floor-admissible class, and $$Q_{1,m}(n)=\lfloor \frac{1+\sqrt{1+8(n-m)}}{2}\rfloor \sim \sqrt{2n}.$$ For $0<\hat{I}\pm <1$, the sequence is identically one. Near criticality, with $\hat{I}\pm =1+\hat{I}´$ and $0<\hat{I}´<(2m-1{)}^{-1}$, we determine the exact first departure time from the critical orbit, of order $\hat{I}{´}^{-2}$. We also prove a finite-step breakdown criterion for large $\hat{I}\pm$ and a conditional slope theorem: any globally defined solution with a limiting density in $(0,1)$ must have slope $1-\hat{I}{\pm }^{-1}$. Exact-arithmetic computations support, but do not prove, a supercritical linear-growth regime.

Major revision, 14 pages, 1 figure. Added a general floor-admissible aggregator invariance theorem, an explicit maximum-aggregator block law, an exact near-critical departure theorem, cross-m exact-arithmetic computations, and a reproducibility archive. Corrected the power-mean numerical conclusions of v1